Oracle SQL, getting a value and using it in multiple case statements

You can put the joins and initial case expression in an inline view or CTE (common table expression, a.k.a subquery factoring), and then query that - you can then refer to the case expression's column alias:

WITH CTE AS (
  SELECT
  r.TRANSACTION_ID,
  CASE
  WHEN TRANSACTION_TYPE = 'RECEIVE'
    THEN r.PRIMARY_QUANTITY * l.UNIT_PRICE
  WHEN TRANSACTION_TYPE = 'RETURN TO VENDOR'
    THEN  -1 * (r.PRIMARY_QUANTITY * l.UNIT_PRICE) 
  END AS "Total",
  glcc.SEGMENT5 AS BinNumber,
  CAST(SUBSTR(d.ATTRIBUTE2, -2) AS INT) AS LineNumber,  
   CASE
        WHEN r.PRIMARY_QUANTITY * l.UNIT_PRICE = aila.AMOUNT
        THEN aila.ACCOUNTING_DATE
        ELSE (SELECT MAX(ACCOUNTING_DATE) FROM AP_INVOICE_LINES_ALL WHERE d.PO_DISTRIBUTION_ID = PO_DISTRIBUTION_ID)
    END AS TransactionDate,
    s.VENDOR_ID AS VendorId,
    s.VENDOR_NAME AS VendorName,
    h.SEGMENT1 AS PONumber,
    l.LINE_NUM AS POLineNumber
  FROM
      RCV_TRANSACTIONS r 
  JOIN
      PO_DISTRIBUTIONS_ALL d 
      ON r.PO_LINE_ID = d.PO_LINE_ID
  JOIN
      GL_CODE_COMBINATIONS glcc 
      ON d.CODE_COMBINATION_ID = glcc.CODE_COMBINATION_ID
  LEFT OUTER JOIN
      AP_SUPPLIERS s
      ON s.VENDOR_ID = r.VENDOR_ID
  LEFT OUTER JOIN
      PO_HEADERS_ALL h
      ON h.PO_HEADER_ID = d.PO_HEADER_ID
  LEFT OUTER JOIN
      PO.PO_LINES_ALL l
      ON l.PO_LINE_ID = r.PO_LINE_ID
  LEFT OUTER JOIN
    AP_INVOICE_LINES_ALL aila
    ON d.PO_DISTRIBUTION_ID = aila.PO_DISTRIBUTION_ID
    AND r.PRIMARY_QUANTITY * l.UNIT_PRICE = aila.AMOUNT
  WHERE
      (TRANSACTION_TYPE = 'RECEIVE' or TRANSACTION_TYPE = 'RETURN TO VENDOR') 
  AND
      (glcc.SEGMENT2 = '1070'
      OR
      glcc.SEGMENT2 = '1071')
)
SELECT
   TRANSACTION_ID,
   "Total",
   BinNumber,
   LineNumber,
   TransactionDate,
   CASE
      WHEN EXTRACT(MONTH FROM TransactionDate) >= 9
      THEN EXTRACT(MONTH FROM TransactionDate) - 8
      ELSE
      EXTRACT(MONTH FROM TransactionDate) + 4
   END AS Period,
   CASE
      WHEN EXTRACT(MONTH FROM TransactionDate) >= 9
      THEN EXTRACT(YEAR FROM TransactionDateE) + 1
      ELSE
      EXTRACT(YEAR FROM TransactionDate)
   END AS BudgetYear,
   EXTRACT(MONTH FROM TransactionDate) AS CalendarMonth,
   EXTRACT(YEAR FROM TransactionDate) AS CalendarYear,
   VendorId,
   VendorName,
   PONumber,
   POLineNumber
  FROM
    CTE;

So the CTE query is everything from your initial query minus the four expressions that referred to aila.ACCOUNTING_DATE. And the query against CTE gets all the columns from that and adds the expressions for those four, but now referring to TransactionDate instead.